Let $f \colon \mathbb{R}\to \mathbb{R}$ be real analytic and let $A\subseteq \mathbb{R}$ be such that the set $A'$ of all accomulation points od $A$ is not empty. If $f(a)=0$ for all $a \in A$ is then necessary $f(t)=0$ for all $t \in \mathbb{R}$?
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2 Answers
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answered Aug 27, 2014 at 19:34
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$\begingroup$ I always thought this was called the "unicity theorem". $\endgroup$ Commented Aug 27, 2014 at 19:38
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1$\begingroup$ @GHfromMO, I think people "believed in" a "principle of permanence" for some time before the notion of holomorphic function was made explicit. And, more often these days I would call it "the identity principle"... $\endgroup$ Commented Aug 27, 2014 at 21:33
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$f$ has a holomorphic extension to a neighborhood of $\mathbb{R}$, so the answer is yes by the unicity theorem.
answered Aug 27, 2014 at 19:30